Automorphisms fixing a variable of K〈x, y

We study automorphisms φ of the free associative algebra K〈x, y, z〉 over a field K such that φ(x), φ(y) are linear with respect to x, y and φ(z) = z. We prove that some of these automorphisms are wild in the class of all automorphisms fixing z, including the well known automorphism discovered by Anick, and show how to recognize the wild ones. This class of automorphisms induces tame automorphisms of the polynomial algebra K[x,y, z]. For n > 2 the automorphisms of K〈x1, . . . , xn, z〉 which fix z and are linear in the xis are tame. Introduction Let K be a field of any characteristic and let X = {x1, . . . , xn}, n ≥ 2, be a finite set. We denote by K[X ] the polynomial algebra in the set of variables X and by K〈X〉 the free associative algebra (or the algebra of polynomials in the set X of noncommuting variables). We write the automorphisms of K[X ] and K〈X〉 as n-tuples of the images of the coordinates, i.e., φ = (f1, . . . , fn) means that φ(xj) = fj(X) = fj(x1, . . . , xn), j = 1, . . . , n. We distinguish two kinds of K-algebra automorphisms of K[X ] and K〈X〉. The first kind are the affine automorphisms